Last year I took a course at UNC with (my homeboy) Zlatko Plese - so I'm not an expert - on variants of early Christianity. One of the chief lessons I learned from it was that in this period where the Roman army and Christianity overlapped, there was a lot of confusion as to whether or not that was -to borrow a Jewish term- kosher. Different Christian religion authorities in the mid-late Empire differed on the subject of Christian participation in the Roman state apparatus. My term paper was on the evolution of Just War within the early Christian intelligentsia, as it were, and at least within that context it took Constantine's "conversion to Christianity" for the Roman military to be an explicitly sanctioned activity among the more orthodox clergy.

(unfortunately I'm on my laptop, so I don't have access to the paper in particular)

To add more detail, the Constantinian "conversion" was explained by men like Eusebius of Caesarea in an almost apocalyptic fervor - the final Christian King to redeem the centuries of persecutions, all of his activities sanctioned (including the civil war against relatively Christian-tolerant rivals) in a language of "us Christians against the devil". Constantine's rivals are described as beholden to sorcerers and daemons.

The truth is, it's doubtful whether the Roman army ever became a "Christian" institution - but its orthodox mandate by mainstream Christian theologians and thinkers largely began with the endorsement of Christianity by Constantine. Demographics of the army aren't largely available, but the opinion of Christian thinkers is available. I know that Christian thinkers argued about it going back to the 2nd century.

Wether or not the Roman Army became a strictly Christian Institution is a difficult to say. Instead we can draw some comparisons and get an idea from the office of the imperial seat. Constantine the Great was the first, and inevitably a lot of his reign is focused on what he did for Christianity. He gave back funds, and church grounds that had been revoked by Diocletian after his persecutions, and generally made life easier for Christians. The last hundreds of years had shown that Christianity was not simply a cult like any others. it was a growing minority that unlike pagans sought to convert people to their religion, and they were also an organized group unlike Roman Paganism. Many times throughouts the Roman Empires history efforts had been made to eradicate them with little success. By Constantines they had become accepted and people understood they were here to stay.

What's notable is that after Constantine there was only ever one Pagan empeoror, Julian the Apostate who tried to revive Roman Paganism and organize it into a single "church" like the Christian community was organized. This failed and persecutions of Pagans were a fact in the last years of the empire. With time dissapeared also the ancient long practice of proclaming late emperors as gods, and rather see them as Gods representative on earth. When Priscus Attalus on Alarics orders were proclaimed Emperor and tried to usurp the throne of Honorius, although he was a pagan he quickly agreed to be baptised, as it was unlikely to be much support for a non christian Emperor. This shows how much the society, institutions and the empire itself had changed with regards to religion from the days of persections of Diocletian.

With regards to the Roman Army, the Roman Empire in general at least remained a place where an individual was free to practice whatever religion they liked even though persecutions throughout history had shown differently to certain people. But as such any individual in the Roman Army was free to practice their own religion, Jews, Christians and Pagans alike, as long as they all swore oaths to the Emperor.

By the time Theodosius made Christianity a state religion in the 390's much Roman society had been Christian for a time. The real development came between the reign of Constantine and Theodosius.

This is also true of the Roman Army although most of the changes that were made to it had nothing to do with Christianity and more to do with other factors. Although it wasn't exactly the answer to your question, I hope it helped.

I apologize for any typos.

Source: The Fall of the West: The Death of the Roman Superpower Adrian Goldsworthy